On 5/22/2012 10:56 AM, Joseph Knight wrote:

On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

    On 5/21/2012 6:26 PM, Russell Standish wrote:
    Hi Russell,

        I once thought that consistency, in mathematics, was the
    indication of existence but situations like this make that idea a
    point of contention... CH and AoC
    <http://en.wikipedia.org/wiki/Axiom_of_choice> are two axioms
    associated with ZF set theory that have lead some people
    (including me) to consider a wider interpretation of mathematics.
    What if all possible consistent mathematical theories must somehow

Joel David Hamkins introduced the "set-theoretic multiverse" idea (link <http://arxiv.org/abs/1108.4223>). The abstract reads:

"The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for."

 Hi Joseph,

Thank you for this comment and link! Do you think that there is a possibility of an "invariance theory", like Special relativity but for mathematics, at the end of this chain of reasoning? My thinking is that any form of consciousness or theory of knowledge has to assume that there is something meaningful to the idea that knowledge implies agency <http://en.wikipedia.org/wiki/Agency_%28philosophy%29> and intention <http://plato.stanford.edu/entries/intention/>...

    Its one reason why Bruno would like to restrict ontology to machines,
    or at most integers - echoing Kronecker's quotable "God made the
    integers, all else is the work of man".

        I understand that, but this choice to restrict makes Bruno's
    Idealism even more perplexing to me; how is it that the Integers
    are given such special status, especially when we cast aside all
    possibility (within our ontology) of the "reality" of the physical
    world? Without the physical world to act as a "selection"
    mechanism for what is "Real", why the bias for integers? This has
    been a question that I have tried to get answered to no avail.

I think Bruno gives such high status to the natural numbers because they are perhaps the least-doubt-able mathematical entities there are. The very fact that talks of a "set-theoretic multiverse" exist makes one ask, how real are sets? Do set theories tell us more about our minds than they do about the mathematical world? (Obviously, as David Lewis pointed out, you need something like a set theory in order to do mathematics at all, and as Russell says, for the average mathematician it really doesn't matter.)

My skeptisism centers on the ambiguity of the metric that defines "the least-doubt-able mathematical entities there are". We operate as if there is a clear domain of meaning to this phrase and yet are free to range outside it at will without self-contradiction. Set theory, whether implicit of explicitly acknowledged seems to be a requirement for communication of the 1st person content. Is it necessary for consciousness itself? Might consciousness, boiled down to its essence, be the act of making a distinction itself?

Also: *No one here has questioned the reality of the physical world. *Should I append this statement to every email until you stop countering it?

I frankly have to explicitly mention this because the "reality of the physical world" is, in fact, being questioned by many posters on this list. That you would write this remark is puzzling to me. I think that I can safely assume that you have read Bruno's papers... Maybe the problem is that I fail to see how reducing the physical world to the epiphenomena of numbers does not also remove its "reality".

    This is the origin of Bruno's claim that COMP entails that physics is
    not computable, a corrolory of which is that Digital Physics is
    refuted (since DP=>COMP).

         Does the symbol "=>" mean "implies"? I get confused ...

    Yes, that is the usual meaning. It can also be written (DP or not COMP).

        "=>" = "or not"]

Actually "a implies b" is defined as "not a or b".

Thank you for this clarification! Would you care to elaborate on this definition?



"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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